Bolzano’s non-monotonic notion of logical consequence Tarski (, ) on consequence relation Carnap () on consequence as informational inclusion
Carnap’s notion of informational containment
and monotonicity
An unfinished sketch
Berislav Žarnić
University of Split
Bolzano’s non-monotonic notion of logical consequence Tarski (, ) on consequence relation Carnap () on consequence as informational inclusion
Overview
 Bolzano’s non-monotonic notion of logical consequence
 Tarski (, ) on consequence relation
Axiom 
The historical significance?
 Carnap () on consequence as informational inclusion
Is Carnap’s consequence monotonic?
Two readings for the information containment notion of consequence
 Consequence relation in practical logic
Geach () and Davidson () on non-monotonic character of
consequence relation in practical logic
 A tentative conclusion
Bolzano’s non-monotonic notion of logical consequence Tarski (, ) on consequence relation Carnap () on consequence as informational inclusion
Introduction
The idea that logic is about just one notion of ‘logical consequence’ is actually
one very particular historical stance. It was absent in the work of the great
pioneer Bernard Bolzano, who thought that logic should chart the many
different consequence relations that we have, depending on the reasoning task
at hand.
van Benthem, J.
() Logic and reasoning: Do the facts matter?.
Studia Logica :–.
Bolzano’s non-monotonic notion of logical consequence Tarski (, ) on consequence relation Carnap () on consequence as informational inclusion
Bolzano’s consequence relation: Consistency and
invariance under substitution
The two conditions:
(a) A, B, C, D,
. . . ,M, N, O, . . . must be
compatible.
(b) Every substitution of
“variable ideas” which makes
all A, B, C, D, . . . true also
makes all M, N, O, . . . true.
The requirement of
compatibility shows that
Bolzano’s consequence
relation is non-monotonic.
For example, p ∈ Cn({p})
but p ∉ Cn({p, ¬p})
Bolzano, B.
([] ) The Theory of Science (Die Wissenschaftslehre oder
Versuch einer Neuen Darstellung der Logik), ed. and translated by
Rolf George.
University of California Press
Bolzano’s non-monotonic notion of logical consequence Tarski (, ) on consequence relation Carnap () on consequence as informational inclusion
Tarski’s consequence
• In . in a lecture in Warsaw and in the paper of . Tarski
introduced a general perspective on consequence relation.
• It is axiomatized by ten axioms divided into two groups:
 Properties of the relation between the sets of sentences considered as
wholes. Nowadays called structural rules.
 Properties of the relation defined in terms of the syntactical properties of
elements in the sets. Nowadays called logical rules.
Bolzano’s non-monotonic notion of logical consequence Tarski (, ) on consequence relation Carnap () on consequence as informational inclusion
Structural properties of consequence relation
Axioms on consequence relation
Axiom . ∣S∣ ≤ ℵ .
Axiom . If X ⊆ S, then X ⊆ Cn(X) ⊆ S.
Axiom . If X ⊆ S, then Cn(Cn(X)) = Cn(X).
Axiom . If X ⊆ S, then Cn(X) =
Cn(Y).
⋃
Y⊆X and ∣Y∣<ℵ
Axiom . There exists a sentence x ∈ S such that
Cn({x}) = S.
Tarski, A.
On some fundamental concepts of metamathematics.
In Logic, semantics, metamathematics : papers from  to .
Clarendon Press, Oxford, , pp. –. First published in
. as
Über einige fundamentale Begriffe der Metamathematik
Comptes Rendus des séances de la Société des Sciences et des
Lettres de Varsovie vol. .: –
For countable languages S
(Axiom ) it holds that:
– consequences of sentences
remain within the same
language and premises are
their own consequences
(reflexivity; Axiom ),
– consequences of
consequences of a set are
already consequences of that
set (transitivity; Axiom ),
– consequences of a set X do
not exceed the consequences
of their finite subsets Y,
which are retained in their
superset X consequences
(compactness, Axiom ),
– there is at least one sentence
in the language such that its
consequences include all the
sentences of that language
(existence of falsum,
“absurdity,” “explosive
sentence,” “informational
breakdown,” etc.; Axiom ).
Bolzano’s non-monotonic notion of logical consequence Tarski (, ) on consequence relation Carnap () on consequence as informational inclusion
Axiom 
Axiom : compactness and restricted monotonicity
If X ⊆ S, then Cn(X) =
⋃
Y⊆X and ∣Y∣<ℵ
Cn(Y). The Axiom . can be rewritten as
the conjunction of the two sentences:


L-R ∀p(p ∈ Cn(X) → ∃Y(∣Y∣ < ℵ ∧ Y ⊆ X ∧ p ∈ Cn(Y))) and
R-L ∀p(∃Y(∣Y∣ < ℵ ∧ Y ⊆ X ∧ p ∈ Cn(Y)) → p ∈ Cn(X)).
The L-R part captures the notion of ‘proofs as finite objects (texts)’. It is related
to ‘compactness theorem’. The R-L part implies monotonicity of the
consequence relation only for finite sets.
Bolzano’s non-monotonic notion of logical consequence Tarski (, ) on consequence relation Carnap () on consequence as informational inclusion
Axiom 
Monotonicity
Theorem
If X ⊆ Y ⊆ S, then Cn(X) ⊆ Cn(Y).
Tarski omits the proof because of its simplicity. The proof requires the use of
both directions of Axiom .
Proof.
Let X ⊆ Y. Assume p ∈ Cn(X). By Axiom . L-R,
∃Z(∣Z∣ < ℵ ∧ Z ⊆ X ∧ p ∈ Cn(Z))). Since inclusion relation is transitive,
Z ⊆ Y. Thus the three conditions are satisfied: Z is a finite subset of Y and p
belongs to Cn(Z). By Axiom . R-L, p ∈ Cn(Y).
Bolzano’s non-monotonic notion of logical consequence Tarski (, ) on consequence relation Carnap () on consequence as informational inclusion
The historical significance?
The historical significance
• Tarski’s axioms introduced a bird’s-eye view on consequence relation.
• Once a structure of consequence relation has been recognized it has
become possible to discover structures of other, weaker consequence-like
relations in language.
• We proceed to Carnap’s semantical notion of consequence as
informational containment and propose the hypothesis that it enables
introduction of a weak, non-monotonic type of consequence relation.
Bolzano’s non-monotonic notion of logical consequence Tarski (, ) on consequence relation Carnap () on consequence as informational inclusion
Possible states of affairs as building blocks for the concept
of information
Content
A possible state of affairs of all objects dealt with in a system S with respect to
all properties and relations dealt with in S is called an L-state with respect to S.
A sentence or sentential class designating an L-state is called a
state-description.
Carnap, R.,
() Introduction to Semantics.
Harvard University Press
Possible
state of
affairs
w
...
w
...
w
Pa
Pb
t
t
Qa
Qb
t
t
t
t
f
f
...
t
f
...
f
f
State description
{Pa, Pb, Qa, Qb}
...
{Pa, ¬Pb, Qa, Qb}
...
{¬Pa, ¬Pb, ¬Qa, ¬Qb}
Bolzano’s non-monotonic notion of logical consequence Tarski (, ) on consequence relation Carnap () on consequence as informational inclusion
The logical range of a sentence
• Carnap explicates the ‘L-range of a sentence S’ in semantic terms as:
class of possible states of affairs “admitted by S”.
• Carnap states the postulates on the parallelism between the inclusion of
logical ranges and the consequence relation.
Postulates for L-range
+P-. If Si → Sj (in S), then LrSi ⊆ LrSi .
L
+P-. If LrSi ⊆ LrSj (in S), then Si → Sj .
L
Carnap, R.,
() Introduction to Semantics.
Harvard University Press
Bolzano’s non-monotonic notion of logical consequence Tarski (, ) on consequence relation Carnap () on consequence as informational inclusion
Consequence and range inclusion
• Tarski’s consequence operation and Carnap’s logical range inclusion can
be related as follows:
Carnap
Lrφ ⊆ Lrψ
Tarski
ψ ∈ Cn({φ})
• The ‘L-range of a sentence S’ can be formulated in the modern notation
as follows (with W standing for the the class of all possible states of affairs
in a given semantical system):
Definition
Lrφ = {w ∈ W ∣ w(φ) = t}
()
LrT = {w ∈ W ∣ w(φ) = t for all φ ∈ T}
()
• The aforementioned postulates, which state that L-range inclusion
parallels consequence relation, can be proved using the definition.
Bolzano’s non-monotonic notion of logical consequence Tarski (, ) on consequence relation Carnap () on consequence as informational inclusion
Is Carnap’s consequence monotonic?
Is Carnap’s consequence monotonic?
• The Tarskian properties of consequence relation can be restated using the
notion of logical range and proved using the definition. The symbol for
the “semantical system” will be omitted.
• Carnap did not explicitly state the theorem on the monotonicity of
consequence relation as based on “logical range” inclusion.
• Tarski’s theorem: If X ⊆ Y, then Cn(X) ⊆ Cn(Y).
• Tarski’s theorem reformulated: If X ⊆ Y, then for all φ if φ ∈ Cn(X), then
φ ∈ Cn(Y).
• Corresponding Carnap’s theorem would be: If X ⊆ Y, then for all φ if
LrX ⊆ Lrφ, then LrY ⊆ Lrφ.
Bolzano’s non-monotonic notion of logical consequence Tarski (, ) on consequence relation Carnap () on consequence as informational inclusion
Is Carnap’s consequence monotonic?
Monotonicity of logical range inclusion
Theorem
If X ⊆ Y, then for all φ if LrX ⊆ Lrφ, then LrY ⊆ Lrφ.
Proof.
Let X ⊆ Y. Suppose LrX ⊆ Lrφ. Let w be an arbitrary state of affairs such that
w ∈ LrY. Since Y = X ∪ (Y − X), w makes true all sentences in X. Therefore,
w ∈ LrX. Given that LrX ⊆ Lrφ, it follows that w ∈ Lrφ.
Bolzano’s non-monotonic notion of logical consequence Tarski (, ) on consequence relation Carnap () on consequence as informational inclusion
Is Carnap’s consequence monotonic?
Logical content as the dual notion of logical range
• For the concept of L-range there is the dual notion of L-content:
LcX = W − LrX.
• Cranap prefers the notion of “logical content” since it accords better with
what is the common use of expressions for relations of relative
informativeness. For example, Pa ∧ Qa is more informative then Pa, but
∣Lr(Pa ∧ Qa∣ < ∣Lr(Pa)∣. On the other hand, the relation between “logical
contents” gives what is desired: the more informative sentence has the
greater cardinality of the logical content. For example,
∣Lc(Pa ∧ Qa)∣ > ∣Lc(Pa)∣.
Bolzano’s non-monotonic notion of logical consequence Tarski (, ) on consequence relation Carnap () on consequence as informational inclusion
Is Carnap’s consequence monotonic?
Information containment conception of logical
consequence
The notion ‘logical content of sentence’ () reappears in . as the notion
of ‘semantic information carried by a sentence’
Information
Whenever i L-implies j, i asserts all that is asserted by j, and possibly more. In
other words, the information carried by i includes the information carried by j
as a (perhaps improper) part. Using ‘In(...)’ as an abbreviation for the
presystematic concept ‘the information carried by . . . ’, we can now state the
requirement in the following way:
R-. In(i) includes In(j) iff i L-implies j.
By this requirement we have committed ourselves to treat information as a set
or class of something. This stands in good agreement with common ways of
expression,as for example, “The information supplied by this statement is more
inclusive than (or is identical with, or overlaps) that supplied by the other
statement.”
Rudolf Carnap and Yehoshua Bar-Hillel.
An Outline of a Theory of Semantic Information.
Technical Report no. . Research Laboratory of Electronics, Massachusetts Institute of
Technology, .
Bolzano’s non-monotonic notion of logical consequence Tarski (, ) on consequence relation Carnap () on consequence as informational inclusion
Is Carnap’s consequence monotonic?
Information containment
Carnap’s explication of consequence relation in terms of relations between
logical contents seems to be the earliest formulation of the “information
containment conception” of consequence relation.
An example
The information containment conception: P implies c if and only if then
information of c is contained in the information of P. In this sense, if P implies
c, then it would be redundant to assert c in a context where the propositions in
P have already been asserted; i.e., no information would be added by asserting
c.
Jose M. Saguillo.
Logical Consequence Revisited. The Bulletin of Symbolic Logic () : -
Bolzano’s non-monotonic notion of logical consequence Tarski (, ) on consequence relation Carnap () on consequence as informational inclusion
Two readings for the information containment notion of consequence
Two readings for the information containment notion of
consequence
• The two notions of “information containment” can be derived from the
Carnap’s explication of consequence relation in terms of “semantic
information” (logical content, logical range):
Strong inf. cont. Conclusion makes no informational change to any
context that includes all the information contained in
premises.
Weak inf. cont. Conclusion makes no informational change to the
context that includes only the information contained in
premises.
• The weak informational containment is a special case of the strong
informational containment.
Bolzano’s non-monotonic notion of logical consequence Tarski (, ) on consequence relation Carnap () on consequence as informational inclusion
Two readings for the information containment notion of consequence
Is information addition the only sentential operation?
The semantics of sentences can be conceived in terms of an operation
performed on the logical content, for example, as “addition of information”.
The two notions coincide in the case of “information addition” conceived as
intersection operation, Lrφ ∩ Lrψ, because of monotonicity (if Lrφ ⊆ Lrφ′ ,
then Lrφ ∩ Lrψ ⊆ Lrφ′ ). On the other hand, if the repertoire of semantic
operations includes operations which test some property of the “logical
content”, like testing whether a certain sentence can be consistently added to it
(which can be compared to the first operational step in Bolzano-type
consequence), then the difference between the two notions will become visible.
If there is a sentence type whose semantic operation is not the “addition of
information” but the “test of possibility of informational addition”, then the
informational content of the sentence type will be context dependent.

Lr(?ψ) = Lrφ  ∩ ⋅ ⋅ ⋅ ∩ Lrφ n in the context Lrφ  ∩ ⋅ ⋅ ⋅ ∩ Lrφ n if
Lrφ  ∩ ⋅ ⋅ ⋅ ∩ Lrφ n ∩ Lrψ ≠ ∅.

Lr(?ψ) = ∅ in the context Lrφ  ∩ ⋅ ⋅ ⋅ ∩ Lrφ n if Lrφ  ∩ ⋅ ⋅ ⋅ ∩ Lrφ n ∩ Lrψ = ∅.
Bolzano’s non-monotonic notion of logical consequence Tarski (, ) on consequence relation Carnap () on consequence as informational inclusion
Practical inference
• The logical theory of practical inference has not as yet reached a clear cut
form. There is no generally known theory that serves as a point of
reference either for the critique or for the further development. The
theoretical reason for the underdevelopment lies in the logical complexity
of practical inference.




The number of logical elements (usually treated as modal operators)
involved in the practical inference is high and their theories are under
dispute.
The consequence relation of practical inference seems not to be Tarskian,
but rather a very weak relation (non-transitive, non-reflexive,
non-monotonic).
There is a rich variety of candidate means-end relations (sufficient,
necessary, INUS, SUIN,... conditions) and this fact produces different forms
of the instrumental type of practical inference.
There seems to be an agreement that practical inference is defeasible or
non-monotonic.
Bolzano’s non-monotonic notion of logical consequence Tarski (, ) on consequence relation Carnap () on consequence as informational inclusion
Geach () and Davidson () on non-monotonic character of consequence relation in practical logic
Geach’s description of the properties of consequence
relation in practical inference
Quotation
Some years ago I read a letter in a political weekly to some such effect as this. ‘I
do not dispute Col. Bogey’s premises, nor the logic of his inference. But even if
a conclusion is validly drawn from acceptable premises, we are not obliged to
accept it if those premises are incomplete; and unfortunately there is a vital
premise missing from the Colonel’s argument-the existence of Communist
China.’ I do not know what Col. Bogey’s original argument had been; whether
this criticism of it could be apt depends on whether it was a piece of indicative
or of practical reasoning. Indicative reasoning from a set of premises, if valid,
could of course not be invalidated because there is a premise “missing” from
the set. But a piece of practical reasoning from a set of premises can be
invalidated thus: your opponent produces a fiat you have to accept, and the
addition of this to the fiats you have already accepted yields a combination
with which your conclusion is inconsistent.
Peter Geach.
Dr. Kenny on practical inference.
Analysis () : –
Bolzano’s non-monotonic notion of logical consequence Tarski (, ) on consequence relation Carnap () on consequence as informational inclusion
Geach () and Davidson () on non-monotonic character of consequence relation in practical logic
Defeasibility of conclusion and completeness of premises
The consequence relation described by Geach has two notable properties:
• (“locality”) conclusion holds in virtue of premises but it can be defeated
by additional premises;
• (existence of the limit) if the premises are complete the conclusion cannot
be defeated (where ‘conclusion is defeated’ means ‘premises are
acceptable and conclusion is not acceptable’).
• The methodology used by Geach shows that in building the theory of
practical inference one can start from the pretheoretical understanding of
the specific consequence relation.
• The locality of consequence relation can be explicated using the weak
notion of informational containment.
Bolzano’s non-monotonic notion of logical consequence Tarski (, ) on consequence relation Carnap () on consequence as informational inclusion
Geach () and Davidson () on non-monotonic character of consequence relation in practical logic
Prima facie consequence
• In a similar manner, Davidson writes that in practical inference one
cannot “detach conclusions about what is desirable (or better) or
obligatory from the principles that lend those conclusions colour”. He
uses the term ‘prima facie’ for this kind of consequence relation. In his
representation of practical inference the conclusion is explicitly
“localized” and written as follows:
prima facie(Conclusion, {Premise , . . . , Premisen })
Davidson, D..
How is weakness of the will possible? ().
In Essays on Actions and Events (Second ed.). Oxford: Clarendon Press.
• Davidson describes a kind of weak consequence relation that is context
dependent.
Bolzano’s non-monotonic notion of logical consequence Tarski (, ) on consequence relation Carnap () on consequence as informational inclusion
Concluding remarks
• The selected results within historical development of the understanding
of logical consequence show that the notions introduced at one stage
need not be preserved at the other although they enable the formulation
of alternative notions and the introduction of new notions.
• Tarski’s theses on structural properties of consequence relation have not
been preserved within the field of practical logic but Tarski’s axioms enabled
recognition of other types of consequence relation.
• Carnap’s informational containment notion of logical consequence enabled
the subsequent development of the “calculus of the informational content”
(for example, in “update semantics”) although Carnap’s notion has not been
preserved in its original form.
• Carnap’s informational containment notion of logical consequence in its
weak variant gives a way of understanding the special character of
consequence relation in practical logic.
Bolzano’s non-monotonic notion of logical consequence Tarski (, ) on consequence relation Carnap () on consequence as informational inclusion
Finiteness
• Axiom . reveals that the syntactic way of
thinking provides the ground for
structural axioms formulation.
• Let us consider an example! Suppose that
the number of objects in the domain is
enumerably infinite and that the name for
each object appears on the list:
a , a , . . . , an , . . . . Assume that for each
i ∈ N it holds that Pai . Obviously one
would want to have (a)
∀xPx ∈ Cn({Pa , . . . , Pan , . . . }), and
would not like to have (b) ∀xPx ∈ Cn(X)
for any proper subset X of
{Pa , . . . , Pan , . . . }. Axiom . L-R forbids
(a) because of (b).
• This example shows that that consequence
relation characterized by axioms – does
not “coincide with the common concept”.
Tarski, A.
() On the concept of logical
consequence.
In Logic, semantics, metamathematics :
papers from  to . Clarendon Press,
Oxford, .